\(\int \frac {5-x}{(3+2 x)^4 \sqrt {2+3 x^2}} \, dx\) [243]

Optimal result

Integrand size = 24, antiderivative size = 99 \[ \int \frac {5-x}{(3+2 x)^4 \sqrt {2+3 x^2}} \, dx=-\frac {13 \sqrt {2+3 x^2}}{105 (3+2 x)^3}-\frac {16 \sqrt {2+3 x^2}}{245 (3+2 x)^2}-\frac {10 \sqrt {2+3 x^2}}{343 (3+2 x)}-\frac {57 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{1715 \sqrt {35}} \] Output:

-13/105*(3*x^2+2)^(1/2)/(3+2*x)^3-16/245*(3*x^2+2)^(1/2)/(3+2*x)^2-10*(3*x 
^2+2)^(1/2)/(1029+686*x)-57/60025*35^(1/2)*arctanh(1/35*(4-9*x)*35^(1/2)/( 
3*x^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 0.82 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.79 \[ \int \frac {5-x}{(3+2 x)^4 \sqrt {2+3 x^2}} \, dx=\frac {-\frac {35 \sqrt {2+3 x^2} \left (2995+2472 x+600 x^2\right )}{(3+2 x)^3}+342 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{180075} \] Input:

Integrate[(5 - x)/((3 + 2*x)^4*Sqrt[2 + 3*x^2]),x]
 

Output:

((-35*Sqrt[2 + 3*x^2]*(2995 + 2472*x + 600*x^2))/(3 + 2*x)^3 + 342*Sqrt[35 
]*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]])/180075
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {688, 27, 688, 27, 679, 488, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {5-x}{(2 x+3)^4 \sqrt {3 x^2+2}} \, dx\)

\(\Big \downarrow \) 688

\(\displaystyle -\frac {1}{105} \int -\frac {3 (41-26 x)}{(2 x+3)^3 \sqrt {3 x^2+2}}dx-\frac {13 \sqrt {3 x^2+2}}{105 (2 x+3)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{35} \int \frac {41-26 x}{(2 x+3)^3 \sqrt {3 x^2+2}}dx-\frac {13 \sqrt {3 x^2+2}}{105 (2 x+3)^3}\)

\(\Big \downarrow \) 688

\(\displaystyle \frac {1}{35} \left (-\frac {1}{70} \int -\frac {10 (53-48 x)}{(2 x+3)^2 \sqrt {3 x^2+2}}dx-\frac {16 \sqrt {3 x^2+2}}{7 (2 x+3)^2}\right )-\frac {13 \sqrt {3 x^2+2}}{105 (2 x+3)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{35} \left (\frac {1}{7} \int \frac {53-48 x}{(2 x+3)^2 \sqrt {3 x^2+2}}dx-\frac {16 \sqrt {3 x^2+2}}{7 (2 x+3)^2}\right )-\frac {13 \sqrt {3 x^2+2}}{105 (2 x+3)^3}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {1}{35} \left (\frac {1}{7} \left (\frac {57}{7} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {50 \sqrt {3 x^2+2}}{7 (2 x+3)}\right )-\frac {16 \sqrt {3 x^2+2}}{7 (2 x+3)^2}\right )-\frac {13 \sqrt {3 x^2+2}}{105 (2 x+3)^3}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {1}{35} \left (\frac {1}{7} \left (-\frac {57}{7} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {50 \sqrt {3 x^2+2}}{7 (2 x+3)}\right )-\frac {16 \sqrt {3 x^2+2}}{7 (2 x+3)^2}\right )-\frac {13 \sqrt {3 x^2+2}}{105 (2 x+3)^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{35} \left (\frac {1}{7} \left (-\frac {57 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{7 \sqrt {35}}-\frac {50 \sqrt {3 x^2+2}}{7 (2 x+3)}\right )-\frac {16 \sqrt {3 x^2+2}}{7 (2 x+3)^2}\right )-\frac {13 \sqrt {3 x^2+2}}{105 (2 x+3)^3}\)

Input:

Int[(5 - x)/((3 + 2*x)^4*Sqrt[2 + 3*x^2]),x]
 

Output:

(-13*Sqrt[2 + 3*x^2])/(105*(3 + 2*x)^3) + ((-16*Sqrt[2 + 3*x^2])/(7*(3 + 2 
*x)^2) + ((-50*Sqrt[2 + 3*x^2])/(7*(3 + 2*x)) - (57*ArcTanh[(4 - 9*x)/(Sqr 
t[35]*Sqrt[2 + 3*x^2])])/(7*Sqrt[35]))/7)/35
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 679

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 688

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( 
(m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2))   Int[(d + 
 e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m 
 + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] 
&& (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.71

method	result	size
risch	\(-\frac {1800 x^{4}+7416 x^{3}+10185 x^{2}+4944 x +5990}{5145 \left (2 x +3\right )^{3} \sqrt {3 x^{2}+2}}-\frac {57 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{60025}\)	\(70\)
trager	\(-\frac {\left (600 x^{2}+2472 x +2995\right ) \sqrt {3 x^{2}+2}}{5145 \left (2 x +3\right )^{3}}-\frac {57 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )-35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{60025}\)	\(77\)
default	\(-\frac {13 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{840 \left (x +\frac {3}{2}\right )^{3}}-\frac {4 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{245 \left (x +\frac {3}{2}\right )^{2}}-\frac {5 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{343 \left (x +\frac {3}{2}\right )}-\frac {57 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{60025}\)	\(95\)

Input:

int((5-x)/(2*x+3)^4/(3*x^2+2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/5145*(1800*x^4+7416*x^3+10185*x^2+4944*x+5990)/(2*x+3)^3/(3*x^2+2)^(1/2 
)-57/60025*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^( 
1/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.05 \[ \int \frac {5-x}{(3+2 x)^4 \sqrt {2+3 x^2}} \, dx=\frac {171 \, \sqrt {35} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 70 \, {\left (600 \, x^{2} + 2472 \, x + 2995\right )} \sqrt {3 \, x^{2} + 2}}{360150 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \] Input:

integrate((5-x)/(3+2*x)^4/(3*x^2+2)^(1/2),x, algorithm="fricas")
 

Output:

1/360150*(171*sqrt(35)*(8*x^3 + 36*x^2 + 54*x + 27)*log(-(sqrt(35)*sqrt(3* 
x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 70*(600*x^2 
 + 2472*x + 2995)*sqrt(3*x^2 + 2))/(8*x^3 + 36*x^2 + 54*x + 27)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {5-x}{(3+2 x)^4 \sqrt {2+3 x^2}} \, dx=\text {Timed out} \] Input:

integrate((5-x)/(3+2*x)**4/(3*x**2+2)**(1/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.05 \[ \int \frac {5-x}{(3+2 x)^4 \sqrt {2+3 x^2}} \, dx=\frac {57}{60025} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) - \frac {13 \, \sqrt {3 \, x^{2} + 2}}{105 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {16 \, \sqrt {3 \, x^{2} + 2}}{245 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {10 \, \sqrt {3 \, x^{2} + 2}}{343 \, {\left (2 \, x + 3\right )}} \] Input:

integrate((5-x)/(3+2*x)^4/(3*x^2+2)^(1/2),x, algorithm="maxima")
 

Output:

57/60025*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x 
 + 3)) - 13/105*sqrt(3*x^2 + 2)/(8*x^3 + 36*x^2 + 54*x + 27) - 16/245*sqrt 
(3*x^2 + 2)/(4*x^2 + 12*x + 9) - 10/343*sqrt(3*x^2 + 2)/(2*x + 3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (80) = 160\).

Time = 0.14 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.34 \[ \int \frac {5-x}{(3+2 x)^4 \sqrt {2+3 x^2}} \, dx=\frac {57}{60025} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {\sqrt {3} {\left (38 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 855 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 2250 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 13290 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3448 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 800\right )}}{3430 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{3}} \] Input:

integrate((5-x)/(3+2*x)^4/(3*x^2+2)^(1/2),x, algorithm="giac")
 

Output:

57/60025*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3* 
x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 1/34 
30*sqrt(3)*(38*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 855*(sqrt(3)*x - 
sqrt(3*x^2 + 2))^4 + 2250*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 13290* 
(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3448*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2 
)) - 800)/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3 
*x^2 + 2)) - 2)^3
 

Mupad [B] (verification not implemented)

Time = 6.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.07 \[ \int \frac {5-x}{(3+2 x)^4 \sqrt {2+3 x^2}} \, dx=\frac {57\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{60025}-\frac {57\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{60025}-\frac {5\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{343\,\left (x+\frac {3}{2}\right )}-\frac {4\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{245\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{840\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \] Input:

int(-(x - 5)/((2*x + 3)^4*(3*x^2 + 2)^(1/2)),x)
 

Output:

(57*35^(1/2)*log(x + 3/2))/60025 - (57*35^(1/2)*log(x - (3^(1/2)*35^(1/2)* 
(x^2 + 2/3)^(1/2))/9 - 4/9))/60025 - (5*3^(1/2)*(x^2 + 2/3)^(1/2))/(343*(x 
 + 3/2)) - (4*3^(1/2)*(x^2 + 2/3)^(1/2))/(245*(3*x + x^2 + 9/4)) - (13*3^( 
1/2)*(x^2 + 2/3)^(1/2))/(840*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8))
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.93 \[ \int \frac {5-x}{(3+2 x)^4 \sqrt {2+3 x^2}} \, dx=\frac {-21000 \sqrt {3 x^{2}+2}\, x^{2}-86520 \sqrt {3 x^{2}+2}\, x -104825 \sqrt {3 x^{2}+2}+1368 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{3}+6156 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{2}+9234 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x +4617 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right )-1368 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{3}-6156 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{2}-9234 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x -4617 \sqrt {35}\, \mathrm {log}\left (2 x +3\right )}{1440600 x^{3}+6482700 x^{2}+9724050 x +4862025} \] Input:

int((5-x)/(3+2*x)^4/(3*x^2+2)^(1/2),x)
 

Output:

( - 21000*sqrt(3*x**2 + 2)*x**2 - 86520*sqrt(3*x**2 + 2)*x - 104825*sqrt(3 
*x**2 + 2) + 1368*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**3 + 
 6156*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**2 + 9234*sqrt(3 
5)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x + 4617*sqrt(35)*log(sqrt(3*x 
**2 + 2)*sqrt(35) + 9*x - 4) - 1368*sqrt(35)*log(2*x + 3)*x**3 - 6156*sqrt 
(35)*log(2*x + 3)*x**2 - 9234*sqrt(35)*log(2*x + 3)*x - 4617*sqrt(35)*log( 
2*x + 3))/(180075*(8*x**3 + 36*x**2 + 54*x + 27))